Time domain calculation method of voltage traveling-wave differential protection for VSC-HVDC transmission lines

ABSTRACT

The present invention related to the field of power system protection and control, presents a voltage travelling wave differential protection method considering the influence of frequency-dependent parameters, which provides more accurate and rapid fault identification. The technical scheme of the present invention is as follows: a calculation method of voltage travelling-wave differential protection for VSC-HVDC transmission lines, taking the influence of the frequency-dependent parameters into consideration, the steps are as follows: calculating a characteristic impedance and propagation coefficient of the frequency-dependent transmission line in time domain, collecting voltage and current signals at the both ends of the transmission line and then calculating the forward and backward voltage travelling wave, respectively, comparing a differential value of voltage travelling wave with a preset threshold value to determine whether the internal fault occurs. 
     The present invention is mainly applied in the field of power system protection and control.

TECHNICAL FIELD

The invention relates to the field of power system protection and control, especially to the travelling-wave differential protection of VSC-HVDC transmission lines, and proposes a calculation method of voltage travelling-wave differential protection for VSC-HVDC transmission lines considering the frequency-dependent line parameters.

BACKGROUND OF THE PRESENT INVENTION

Voltage Source Converter Based High Voltage Direct Current (VSC-HVDC) is a novel High Voltage Direct Current (HVDC) based on Voltage Source Converter (VSC) and full-controlled turn-off electronic device. The VSC-HVDC has the advantages such as high economic efficiency, flexibility and environmental friendly protection, and is one of the key techniques in the research and application of smart grid. At present, the VSC-HVDC adopts Modular Multilevel converters (MMC) to convert AC to DC. Due to the small damping at DC side, the current suddenly rises and voltage drops rapidly when a short circuit occurs on the VSC-HVDC transmission line, which will extremely threaten the security of system and device. Therefore, it is important to develop a protection principle of fast, reliable and selective identifying faults so as to ensure the security and stability of the VSC-HVDC transmission systems.

Currently, the protection of the VSC-HVDC transmission lines still follows the conventional transmission line protection strategy of the HVDC system, which uses travelling wave protection as a fast main protection and current differential protection as a backup protection. The conventional longitudinal differential protection has the advantages such as good selectivity and high sensitivity, but also has long time delay and cannot meet the speed requirement of the VSC-HVDC system. Besides, the transient electrical quantities of the VSC-HVDC are not the same as the AC system that take the power frequency as main frequency, but comprise rich frequency components. And the transmission process of these quantities in the DC line is greatly affected by the line frequency-dependent parameters. Therefore, taking the influence of the frequency-dependent line parameters of the VSC-HVDC system into consideration and decreasing time delay of the longitudinal differential protection are the basic premises for the protection principle to be successfully applied to the VSC-HVDC transmission line and developing its inherent advantages.

SUMMARY OF THE PRESENT INVENTION

In order to overcome the defects of the prior art, in view of the fact that the conventional longitudinal differential protection of the VSC-HVDC transmission lines does not take the frequency-dependent line parameters into consideration and the problems that the long time delay is not conducive to the rapid identification of the fault, the present invention provides a voltage travelling-wave differential protection method considering the frequency-dependent characteristic of line parameters. The proposed method of the present invention provides more accurate and rapid detection of the internal fault. The technical scheme of the present invention is as follows: a calculation method of voltage travelling-wave differential protection for VSC-HVDC transmission lines, considering the influence of the frequency-dependent characteristic of line parameters, the steps are as follows: calculating the characteristic impedance and propagation coefficient of the frequency-dependent transmission line in time domain, collecting the voltage and current signals at the both ends of the transmission line and then calculating the forward and backward voltage travelling-wave respectively, comparing the differential value of the above voltage travelling-waves with the preset threshold to determine whether the internal fault occurs.

The detailed steps in an embodiment are as follows:

Step A: calculating a characteristic impedance and a propagation coefficient of the frequency-dependent transmission line in time domain;

Step B: collecting voltage and current signals at the both ends of the transmission line, comprising a head end voltage value u_(m), a head end current value i_(m), a tail end voltage value u_(n), a tail end current value i_(n) of the transmission line, a sampling frequency f_(s), and a number of the sampling voltage and current N_(s); according to the waveform reduction method, filtering and intercepting the voltage and current signals using a filter with kernel length of M₂ and then increasing the sampling frequency f_(s) to a resampling frequency f_(maxres) (f_(maxres)>f_(s)), and the number of sampling points becoming N_(x)−M₂/2, wherein N_(x)=N_(s) f_(maxres)/f_(s); and performing phase mode transformation;

Step C: comparing a differential value of voltage travelling-wave with a preset threshold value so as to determine whether the internal fault occurs:

Calculating differential values of voltage travelling-wave uwave_(diffm) and uwave_(diffn) at both ends, presetting a preset threshold value u_(set), determining whether the internal fault occurs on the transmission line based on the criterion of uwave_(diffm)[n₃]≥u_(set)|uwave_(diffn)[n₃]≥u_(set) (n₃=1, 2 . . . N_(x)−M₂/2), and using the differential-mode components of voltage and current to realize fault recognition; and then further determining whether the internal fault occurs; uwave_(diffm)[n ₃]≥u _(set) ∥uwave_(diffn)[n ₃]≥u _(set)

That is, when either of uwave_(diffm)[n₃] or uwave_(diffn)[n₃] is greater than the preset threshold value u_(set), it is considered that the internal fault occurs at a time t[n₃]. In order to ensure the reliability of the determination, the determination is made continuously several times or by the integral method.

The detailed steps of Step A are as follows:

Step A-1: according to a conductance matrix G and geometric transmission line parameters including a radius of conductor, an average above-ground height of the conductor, a distance between the conductors, a distance between the conductor and the mirror of another conductor, obtaining an impedance matrix of the conductor by the Carson formula, obtaining a capacitance matrix using the potential coefficient matrix, and then calculating the expression of the characteristic impedance and propagation coefficient corresponding to a full length l of the transmission line in frequency domain, and finally transforming it into the time domain by rational function fitting or inverse Fourier transformation;

Step A-2: in order to match the calculated characteristic impedance and propagation coefficient with the collected voltage and current in frequency domain, filtering the characteristic impedance and propagation coefficient by a sinc filter with a Blackman window;

Step A-3: reducing the sampling frequency f_(s) of the filtered characteristic impedance and propagation coefficient corresponding to the full length of the transmission line to the resampling frequency f_(maxres).

The detailed steps of step A-1 are:

-   -   Setting a sampling time t_(max), a sampling frequency f_(max), a         number of sampling points N_(t) in time domain and a number of         sampling points N_(f) in frequency domain as:         N _(t) =N _(f) =t _(max) ·f _(max)

Calculating a sampling time t[n₁] corresponding to the n₁ ^(th) sampling point: t[n ₁]=(n ₁−1)/f _(max)

Where n₁=1, 2, 3 . . . N_(t), calculating a frequency f[n₁] corresponding to the n₁ ^(th) sampling point: f[n ₁]=(n ₁−1)/t _(max)

Step (1): calculating an impedance matrix Z using Carson formula:

Self-impedance Z_(si) (i=1,2) is:

$Z_{si} = {\left( {R_{i,{ac}} + {\Delta R_{si}}} \right) + {j\left( {{\omega\frac{\mu_{0}}{2\pi}\ln\frac{2h_{i}}{r_{i}}} + X_{i,{ac}} + {\Delta X_{si}}} \right)}}$

Where R_(i,ac) is an AC resistance of the conductor i, X_(i,ac) is an AC reactance of the conductor i, ΔR_(si) and ΔX_(si) are Carson's correction terms for earth return effects, h_(i) is an average above-ground height of the conductor i, r_(i) is a radius of the conductor i, μ₀ having the value of 4π×10⁻⁴ H/m is the uniform permeability for both the aerial space and the earth; and angular frequency ω=2πf[n₁];

Mutual impedance Z_(mij) is:

$Z_{mij} = {Z_{mji} = {{\Delta R_{mij}} + {j\left( {{\omega\frac{\mu_{0}}{2\pi}\ln\frac{d_{{ij},{mir}}}{d_{ij}}} + {\Delta X_{mij}}} \right)}}}$

Where j=1,2 and j≠i, d_(ij) is a distance between conductor i and conductor j, d_(ij,mir) is a distance between the conductor i and the mirror of conductor j, ΔR_(mij) and ΔX_(mij) are Carson's correction terms for earth return effects;

$Z = \begin{bmatrix} Z_{s1} & Z_{m12} \\ Z_{m21} & Z_{s2} \end{bmatrix}$

Step (2): calculating a potential coefficient matrix P:

${P_{si} = {\frac{1}{2\pi ɛ_{0}}\ln\frac{2h_{i}}{r_{i}}}}{P_{mij} = {P_{mji} = {\frac{1}{2\pi ɛ_{0}}\ln\frac{d_{{ij},{mir}}}{d_{ij}}}}}$

Where P_(si) is a diagonal element of matrix P, P_(mij) is a off-diagonal element of matrix P, ε₀ is a space permittivity value having the value of 8.85×10⁻¹² F/m, and a capacitance matrix C=P⁻¹;

Step (3): performing phase mode transformation: Z′=SZS ⁻¹ C′=SCS ⁻¹ G′=SGS ⁻¹

Where Z′ is a modulus of the impedance per unit length of the conductor, C′ is a modulus of the capacitance per unit length of the conductor, G′ is a modulus of the conductance per unit length of the conductor, and S is a decoupling matrix,

${S = {\frac{1}{2}\begin{bmatrix} 1 & {- 1} \\ 1 & 1 \end{bmatrix}}};$

Step (4): calculating the function of a characteristic impedance Z_(c) and a propagation coefficient A in frequency domain:

Z^(′) = R^(′) + j ω L^(′) ${Z_{c}(\omega)} = \sqrt{\frac{R^{\prime} + {j\;\omega\; L^{\prime}}}{G^{\prime} + {j\;\omega\; C^{\prime}}}}$ ${A(\omega)} = e^{{- l}\sqrt{{({R^{\prime} + {j\;\omega\; L^{\prime}}})}{({G^{\prime} + {j\;\omega\; C^{\prime}}})}}}$

Where R′ is a resistance per unit length of the conductor, L′ is an inductance per unit length of the conductor;

Step (5): converting the functions from the frequency domain into the time domain using rational function fitting method:

Fitting functions Z_(c)(ω) and A(ω) in the frequency domain by the rational function fitting method, a point where the slope of the fitted line changes is a pole and zero point of the rational function, so the approximate expression of the characteristic impedance and the propagation coefficient in the frequency domain are:

${{Z_{c,{{app}rox}}(s)} = {k\frac{\left( {s + z_{1}} \right)\left( {s + z_{2}} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {s + z_{m1}} \right)}{\left( {s + p_{1}} \right)\left( {s + p_{2}} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {s + p_{m1}} \right)}}}{{A_{approx}(s)} = {e^{{- s}\tau_{\min}}k\frac{\left( {s + z_{1}} \right)\left( {s + z_{2}} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {s + z_{m1}} \right)}{\left( {s + p_{1}} \right)\left( {s + p_{2}} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {s + p_{m2}} \right)}}}$

Where s=jω and m1<m2; z_(m1) is the zero point, m1=1, 2, 3 . . . , P_(m2) is the pole, m2=1, 2, 3 . . . , the zero and pole point are both negative real numbers; k is an coefficient, τ_(min) is a shortest propagation time of the wave; Z_(c,approx)(s) is the rational function approximation of characteristic impedance, A_(approx)(s) is the rational function approximation of propagation coefficient;

Converting the functions into the time domain:

  Z_(c)(t) = k₀δ(t) + k₁e^(−p₁t) + k₂e^(−p₂t) + … + k_(m1)e^(−p_(m1)t) ${a(t)} = \left\{ \begin{matrix} {{{k_{1}e^{- {p_{1}{({t - \tau_{\min}})}}}} + {k_{2}e^{- {p_{2}{({t - \tau_{\min}})}}}} + \ldots + {k_{m\; 2}e^{- {p_{m\; 2}{({t - \tau_{\min}})}}}}},} & {{{if}\mspace{14mu} t} \geq \tau_{\min}} \\ 0 & {{{if}\mspace{14mu} t} < \tau_{\min}} \end{matrix} \right.$

Where Z_(c)(t) is a time domain expression of the characteristic impedance, a(t) is a time domain expression of the propagation coefficient, k_(m2) is a coefficient of the expanded rational function; the characteristic impedance Z_(c)[n₁] and the propagation coefficient a[n₁] corresponding to each sampling time t[n₁] can be achieved.

The detailed steps of Step A-2 are:

Calculating a kernel h[j₁] of a sinc filter with Blackman window:

$\quad\left\{ \begin{matrix} {{h\left\lbrack j_{1} \right\rbrack} = {K\frac{\sin\left( {2\pi\;{f_{c}\left( {j_{1} - {M/2}} \right)}} \right.}{j_{1} - {M/2}}{w\left\lbrack j_{1} \right\rbrack}}} & {j_{1} \neq {M/2}} \\ {{h\left\lbrack j_{1} \right\rbrack} = {2\pi\; f_{c}}} & {j_{1} = {M/2}} \end{matrix} \right.$

-   -   Performing Normalization:

${h\left\lbrack j_{1} \right\rbrack} = \frac{h\left\lbrack j_{1} \right\rbrack}{\sum\limits_{i = 1}^{M + 1}{h\left\lbrack j_{1} \right\rbrack}}$

Where j₁=1, 2 . . . M+1, f_(c) is the cut-off frequency having the value within the range of 0 to 0.5; M is a kernel length of the filter, which should be an even number; K is the coefficient; w[j₁] is the function of Blackman window and is expressed as follows: w[j ₁]=0.42−0.5 cos(2πj ₁ /M)+0.08 cos(4πj ₁ /M)

Performing Filtering

${{z_{cfilter}\left\lbrack {n_{1} + j_{1} - 1} \right\rbrack} = {\sum\limits_{j_{1} = 1}^{M + 1}{{h\left\lbrack j_{1} \right\rbrack}{z_{c}\left\lbrack n_{1} \right\rbrack}}}},{{a_{filter}\left\lbrack {n_{1} + j_{1} - 1} \right\rbrack} = {\sum\limits_{j_{1} = 1}^{M + 1}{{h\left\lbrack j_{1} \right\rbrack}{a\left\lbrack n_{1} \right\rbrack}}}}$

Where n₁=1, 2 . . . N_(t); Z_(cfilter) is a filtered characteristic impedance; a_(filter) is a filtered propagation coefficient; the filtered signal has a delay of M/2 point compared with the original signal, so the filtered point corresponding to the sampling time t[n₁] is Z_(cfilter)[n₁+M/2] and a_(filter)[n₁+M/2].

The detailed steps of Step A-3 are:

Calculating the number of resampling points N_(tres) in the time domain: N _(tres) =t _(max) ·f _(maxres)

Where f_(maxres) is the resampled sampling frequency;

Calculating the sampling time t_(new)[n₂] corresponding to the n₂ ^(th) sampling point: t _(new)[n ₂]=(n ₂−1)/f _(maxres)

n₂=1, 2, 3 . . . N_(tres), and obtaining the resampled sampling value by linear interpolation method:

${z_{cres}\left\lbrack n_{2} \right\rbrack} = {{{\frac{z_{c2} - z_{c1}}{t_{2} - t_{1}} \cdot \left( {{t_{new}\left\lbrack n_{2} \right\rbrack} - t_{1}} \right)} + {z_{c1}{a_{res}\left\lbrack n_{2} \right\rbrack}}} = {{\frac{a_{2} - a_{1}}{t_{2} - t_{1}} \cdot \left( {{t_{new}\left\lbrack n_{2} \right\rbrack} - t_{1}} \right)} + a_{1}}}$

Where Z_(cres) is a resampled characteristic impedance; a_(res) is a resampled propagation coefficient; t₁ and t₂ are two adjacent sampling times in the original sampling time series t, and t₁≤t_(new)[n₂]≤t₂; z_(c1) and z_(c2) are sampling values corresponding to the sampling times t₁, t₂ in the filtered characteristic impedance Z_(cfilter), respectively; a₁ and a₂ are sampling values corresponding to the sampling times t₁, t₂ in the filtered propagation coefficient a_(filter) respectively.

The detailed steps of Step B are as follows:

Step B-1: calculating according to the waveform reduction method:

Step(1): Setting the frequency for traveling-wave transmission calculation as f_(maxres), inserting zero points to an original signal x[i₂] according to the new sampling rate, i₂=1, 2, . . . N_(s), so that the number of sampling points is changed from N_(s) to N_(x), N_(x)=N_(s)·N_(add), N_(add)=f_(maxres)/f_(s), thus obtaining a recombinant signal x_(rec)[i₃]:

-   -   x_(rec)[1+(i₂−1)·N_(dd)]=x[i₂], the values of the rest points         are zero;

Where, i₃=1, 2 . . . N_(x); x represents sampling values including voltages u_(m), u_(n) and currents i_(m), i_(n) at the sampling frequency f_(s); and x_(rec) represents sampling values including voltages u_(mrec), u_(nrec) and currents i_(mrec), i_(nrec) at the sampling frequency f_(max) after inserting the zero points;

Step (2): filtering the recombinant signal x_(rec)[i₃] using the sinc filter with Blackman window, setting a cut-off frequency f_(c2)=1/(2·N_(add)), choosing an appropriate kernel length M₂ and a kernel h₂[j₂] of the filter, and filtering out harmonics at higher frequencies:

${x_{restore}\left\lbrack {i_{3} + j_{2} - 1} \right\rbrack} = {\sum\limits_{j_{2} = 1}^{M_{2} + 1}{{h_{2}\left\lbrack j_{2} \right\rbrack}{x_{rec}\left\lbrack i_{3} \right\rbrack}}}$

Where j₂=1, 2 . . . M₂+1 and i₃=1, 2 . . . N_(x); x_(restore) represents voltage values u_(mrestore), u_(nrestore) and currents values i_(mrestore), i_(nrestore) after waveform reduction;

Step (3): intercepting the restored voltage and current signals, namely, removing the previous and subsequent M₂/2 sampling points of the signal x_(restore); the restored signal is x_(restore)[i₃+M₂/2] corresponding to the sampling time t[i₃], the intercepted signal after waveform reduction is x_(re)[n₃] (n₃=1, 2 . . . N_(x)−M₂/2), and x_(re)[n₃]=x_(restore)[n₃+M₂/2]; x_(re) represents intercepted voltage values u_(mre), U_(nre) and intercepted currents values i_(mre), i_(nre) after waveform reduction.

Step B-2: performing phase mode transformation:

$\begin{bmatrix} u_{{mmod}\; 1} \\ u_{{mmod}\; 2} \end{bmatrix} = {{{\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & {- \frac{\sqrt{2}}{2}} \end{bmatrix}\begin{bmatrix} u_{mp} \\ u_{mn} \end{bmatrix}}\begin{bmatrix} u_{{nmod}\; 1} \\ u_{{nmod}\; 2} \end{bmatrix}} = {{{\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & {- \frac{\sqrt{2}}{2}} \end{bmatrix}\begin{bmatrix} u_{np} \\ u_{nn} \end{bmatrix}}\begin{bmatrix} i_{{mmod}\; 1} \\ i_{{mmod}\; 2} \end{bmatrix}} = {{{\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & {- \frac{\sqrt{2}}{2}} \end{bmatrix}\begin{bmatrix} i_{mp} \\ i_{mn} \end{bmatrix}}\begin{bmatrix} i_{{nmod}\; 1} \\ i_{{nmod}\; 2} \end{bmatrix}} = {\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & {- \frac{\sqrt{2}}{2}} \end{bmatrix}\begin{bmatrix} i_{np} \\ i_{nn} \end{bmatrix}}}}}$

Where the voltage values u_(mmod1), u_(nmod1) and current values i_(mmod1), i_(nmod1) are ground-mode components after waveform reduction; and voltage values u_(mmod2), u_(nmod2) and current values i_(mmod2), i_(nmod2) are differential-mode components after waveform reduction; voltage values u_(mp), u_(mn) are p-phase and n-phase of the voltage value u_(mre), respectively; voltage values u_(np), u_(nn) are p-phase and n-phase of the voltage value u_(nre), respectively; current values i_(mp), i_(mn) are p-phase and n-phase of the current value i_(mre), respectively; current values i_(np), i_(nn) are p-phase and n-phase of the current value i_(nre), respectively.

The detailed steps of Step C are as follows:

Calculating the differential values of voltage travelling-wave at both ends: uwave_(diffm)[n ₃]=(u _(mmod2)[n ₃]+z _(cres)[n ₂]*i _(mmod2)[n ₃])*a _(res)[n ₂]−(u _(nmod2)[n ₃]−z _(cres)[n ₂]*i _(nmod2)[n ₃]) uwave_(diffn)[n ₃]=(u _(nmod2)[n ₃]+z _(cres)[n ₂]*i _(nmod2)[n ₃])*a _(res)[n ₂]−(u _(mmod2)[n ₃]−z _(cres)[n ₂]*i _(mmod2)[n ₃])

Where n₃=1, 2 . . . N M₂/2 and n₂=1, 2 . . . N_(tres); * represents the convolution calculation, the value uwave_(diffn), is the differential value between the differential-mode forward voltage traveling-wave at the head end after full-length line transmission and the differential-mode backward voltage traveling-wave at the tail end; the value uwave_(diffn) is the differential value between the differential-mode forward voltage traveling-wave at the tail end after full-length line transmission and the differential-mode backward voltage traveling-wave at the head end;

Calculating the convolution calculation:

Supposing xx[j₃] is an input signal with N₁ points, j₃ is a positive integer from 1 to N₁, hh[j₄] is an impulse response with N₂ points, and j₄ is a positive integer from 1 to N₂. The convolution result of xx[j₃] and hh[j₄] is y[j₅], which is a signal with N₁+N₂−1 points, j₅ is a positive integer from 1 to N₁+N₂−1, then

${y\left\lbrack j_{5} \right\rbrack} = {\sum\limits_{j_{4} = 1}^{N_{2}}{h{h\left\lbrack j_{4} \right\rbrack}x{x\left\lbrack {j_{5} - j_{4}} \right\rbrack}}}$

Where j₅=1, 2 . . . N₁+N₂−1 and j₄=1, 2 . . . N₂.

The technical features and beneficial advantages of the present invention:

1. Considering the effect of the frequency-dependent parameters during the transmission of transient electrical quantities in the VSC-HVDC, the method of the present invention is realized based on the electrical quantities at double-end of the line which can meet the requirements of selectivity.

2. Compared with the traditional longitudinal differential protection, the method can shorten the delay while ensuring the calculation accuracy, which is beneficial to the rapid identification of faults, and can be used as the main protection when the length of the transmission line is not too long.

3. The method has clear calculation of voltage traveling wave and is easy to implement.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is the schematic diagram of the VSC-HVDC transmission line;

FIG. 2 is the calculation flow chart of the characteristic impedance and propagation coefficient of the transmission line in time domain;

FIG. 3 is the flow chart of the waveform reduction method.

DETAILED DESCRIPTION OF THE PRESENT INVENTION

In order to overcome the defects of the prior art, the present invention provides a calculation method of voltage travelling-wave differential protection for VSC-HVDC transmission lines. The method of the present invention collects the electrical quantity in time domain at the both ends of the transmission line, establishes the frequency-dependent modeling functions of the VSC-HVDC transmission line for calculating the voltage traveling-wave values at both ends, and finally calculates differential value so as to identify whether internal fault occurs. The method can on-line calculate voltage traveling-wave values of the line according to the real-time collected electrical quantity, and can shorten the delay during calculation so as to satisfy the requirements of speed and selectivity of VSC-HVDC system.

The technical scheme of the present invention is as follows: a calculation method of voltage travelling-wave differential protection for VSC-HVDC transmission lines, considering the influence of the frequency-dependent parameters, the steps are as follows: calculating the characteristic impedance and propagation coefficient of the frequency-dependent transmission line in time domain, collecting voltage and current signals at the both ends of the transmission line and then calculating the forward and backward voltage travelling-wave, respectively, comparing the differential value of voltage travelling-wave with the preset threshold value so as to determine whether the internal fault occurs.

An implementation of the present invention will be further described below in detail.

Step A: calculating the characteristic impedance and propagation coefficient of the frequency-dependent transmission line in time domain; the detailed steps are as follows:

Step A-1: according to the conductance matrix G (generally ignored, can be set to 0) and the geometric transmission line parameters including the radius of conductor, the average above-ground height of the conductor, the distance between the conductors, the distance between the conductor and the mirror of another conductor, obtaining the impedance matrix of the conductor by the Carson formula, obtaining the capacitance matrix using the potential coefficient matrix, and then calculating the expression of the characteristic impedance and propagation coefficient corresponding to the full length l of the transmission line in frequency domain, and finally transforming it into the time domain by rational function fitting or inverse Fourier transformation;

Setting the sampling time t_(max), the sampling frequency f_(max), the number of sampling points N_(t) in time domain and the number of sampling points N_(f) in frequency domain as: N _(t) =N _(f) =t _(max) ·f _(max)

Calculating the sampling time t[n₁] (n₁=1, 2, 3 . . . N_(t)) corresponding to the n₁ ^(th) sampling point: t[n ₁]=(n ₁−1)/f _(max)

Calculating the frequency f[n₁] corresponding to the n₁ ^(th) sampling point: f[n ₁]=(n ₁−1)/t _(max)

-   -   a): calculating the impedance matrix Z using Carson formula:     -   Self-impedance Z_(si) (i=1,2) is:

$Z_{sj} = {\left( {R_{i,{ac}} + {\Delta R_{si}}} \right) + {j\left( {{\omega\frac{\mu_{0}}{2\pi}\ln\frac{2h_{i}}{r_{i}}} + X_{i,{ac}} + {\Delta X_{si}}} \right)}}$

-   -   Where R_(i,ac) is the AC resistance of the conductor i, X_(i,ac)         is the AC reactance of the conductor i, ΔR_(si) and ΔX_(si) are         Carson's correction terms for earth return effects, h_(i) is the         average above-ground height of the conductor i, r_(i) is the         radius of the conductor i, μ₀ having the value of 4π×10⁻⁴ H/m is         the uniform permeability for both the aerial space and the         earth; and angular frequency ω=2πf[n₁];     -   Mutual impedance Z_(mij) (j=1,2 and j≠i) is:

$Z_{mij} = {Z_{mji} = {{\Delta R_{mij}} + {j\left( {{\omega\frac{\mu_{0}}{2\pi}\ln\frac{d_{{ij},{mir}}}{d_{ij}}} + {\Delta X_{mij}}} \right)}}}$

-   -   Where d_(ij) is the distance between conductor i and conductor         j, d_(ij,mir) is the distance between the conductor I and the         mirror of conductor j, ΔR_(mij) and ΔX_(mij) are Carson's         correction terms for earth return; and then

${Z = \begin{bmatrix} Z_{s1} & Z_{m12} \\ Z_{m21} & Z_{s2} \end{bmatrix}};$

-   -   b): calculating the potential coefficient matrix P:

${P_{si} = {\frac{1}{2\pi ɛ_{0}}\ln\frac{2h_{i}}{r_{i}}}}{P_{mij} = {P_{mji} = {\frac{1}{2\pi ɛ_{0}}\ln\;\frac{d_{{ij},{mir}}}{d_{ij}}}}}$

Where P_(si) is the diagonal element of matrix P, P_(mij) is the off-diagonal element of matrix P, ε₀ is the space permittivity value having the value of 8.85×10⁻¹² F/m, and the capacitance matrix C=P⁻¹;

-   -   c): performing phase mode transformation:         Z′=SZS ⁻¹         C′=SCS ⁻¹         G′=SGS ⁻¹     -   Where Z′ is the modulus of the impedance per unit length of the         conductor, C′ is the modulus of the capacitance per unit length         of the conductor, G′ is the modulus of the conductance per unit         length of the conductor, and S is the decoupling matrix,

${S = {\frac{1}{2}\begin{bmatrix} 1 & {- 1} \\ 1 & 1 \end{bmatrix}}};$

-   -   d): Calculating the function of the characteristic impedance         Z_(c) and the propagation coefficient A in frequency domain:

Z^(′) = R^(′) + j ω L^(′) ${Z_{c}(\omega)} = \sqrt{\frac{R^{\prime} + {j\;\omega\; L^{\prime}}}{G^{\prime} + {j\;\omega\; C^{\prime}}}}$ ${A(\omega)} = e^{{- l}\sqrt{{({R^{\prime} + {j\;\omega\; L^{\prime}}})}{({G^{\prime} + {j\;\omega\; C^{\prime}}})}}}$

-   -   Where R′ is the resistance per unit length of the conductor, L′         is the inductance per unit length of the conductor;     -   e): converting the functions from the frequency domain into the         time domain using rational function fitting method:     -   Fitting the functions Z_(c)(ω) and A(ω) in the frequency domain         by the rational function fitting method, the point where the         slope of the fitted line changes is the pole and zero point of         the rational function, so the approximate expression of the         characteristic impedance and the propagation coefficient in the         frequency domain are:

${{Z_{c,{{app}rox}}(s)} = {k\frac{\left( {s + z_{1}} \right)\left( {s + z_{2}} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {s + z_{m1}} \right)}{\left( {s + p_{1}} \right)\left( {s + p_{2}} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {s + p_{m1}} \right)}}}{{A_{approx}(s)} = {e^{{- s}\tau_{\min}}k\frac{\left( {s + z_{1}} \right)\left( {s + z_{2}} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {s + z_{m1}} \right)}{\left( {s + p_{1}} \right)\left( {s + p_{2}} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {s + p_{m2}} \right)}}}$

-   -   Where s=jω and m1<m2; Z_(m1) is the zero point, m1=1, 2, 3 . . .         , P_(m2) is the pole, m2=1, 2, 3 . . . , the zero and pole         points are both negative real numbers; k is coefficient, τ_(min)         is the shortest propagation time of the wave; Z_(c,approx)(s) is         the rational function approximation of characteristic impedance,         A_(approx)(s) is the rational function approximation of         propagation coefficient; Converting the functions into time         domain:

Z_(c)(t) = k₀δ(t) + k₁e^(−p₁t) + k₂e^(−p₂t) + … + k_(m1)e^(−p_(m1)t) ${a(t)} = \left\{ {\begin{matrix} {{{k_{1}e^{- {p_{1}{({t - \tau_{\min}})}}}} + {k_{2}e^{- {p_{2}{({t - \tau_{\min}})}}}} + \ldots + {k_{m2}e^{- {p_{m2}{({t - \tau_{\min}})}}}}},} \\ 0 \end{matrix}\begin{matrix} {{{if}\mspace{14mu} t} \geq \tau_{\min}} \\ {{{if}\mspace{14mu} t} < \tau_{\min}} \end{matrix}} \right.$

-   -   Where Z_(c)(t) is the time domain expression of the         characteristic impedance, a(t) is the time domain expression of         the propagation coefficient, k_(m2) is the coefficient of the         expanded rational function; the characteristic impedance         Z_(c)[n₁] and the propagation coefficient and corresponding to         each sampling time t[n₁] can be achieved;

Step A-2: in order to match the calculated characteristic impedance and propagation coefficient with the collected voltage and current in frequency domain, filtering the characteristic impedance and propagation coefficient by a sinc filter with a Blackman window;

-   -   Calculating the kernel h[j₁] of a sinc filter with Blackman         window:

$\left\{ {\begin{matrix} {{h\left\lbrack j_{1} \right\rbrack} = {K\frac{\sin\left( {2\pi\;{f_{c}\left( {j_{1} - {M/2}} \right)}} \right.}{j_{1} - {M/2}}{w\left\lbrack j_{1} \right\rbrack}}} & \left( {j_{1} \neq {M/2}} \right) \\ {{h\left\lbrack j_{1} \right\rbrack} = {2\pi\; f_{c}}} & \left( {j_{1} = {M/2}} \right) \end{matrix}\quad} \right.$

-   -   Performing normalization:

${h\left\lbrack j_{1} \right\rbrack} = \frac{h\left\lbrack j_{1} \right\rbrack}{\underset{i = 1}{\sum\limits^{M + 1}}{h\left\lbrack j_{1} \right\rbrack}}$

-   -   Where j₁=1, 2 . . . M+1, f_(c) is the cut-off frequency having         the value within the range of 0 to 0.5; M is the kernel length         of the filter, which should be an even number; K is the         coefficient; w[j₁] is the function of Blackman window and is         expressed as follows:         w[j ₁]=0.42−0.5 cos(2πj ₁ /M)+0.08 cos(4πj ₁ /M)     -   Performing filtering:

${{z_{cfilter}\left\lbrack {n_{1} + j_{1} - 1} \right\rbrack} = {\sum\limits_{j_{1} = 1}^{M + 1}{{h\left\lbrack j_{i} \right\rbrack}{z_{c}\left\lbrack n_{1} \right\rbrack}}}},{{a_{filter}\left\lbrack {n_{1} + j_{1} - 1} \right\rbrack} = {\sum\limits_{j_{1} = 1}^{M + 1}{{h\left\lbrack j_{i} \right\rbrack}{a\left\lbrack n_{1} \right\rbrack}}}},$

-   -   Where n₁=1, 2 . . . N_(t); Z_(cfilter) is the filtered         characteristic impedance; a_(filter) is the filtered propagation         coefficient; the filtered signal has a delay of M/2 point         compared with the original signal, so the filtered point         corresponding to the sampling time t[n₁] is Z_(cfilter)[n₁+M/2]         and a_(filer)[n₁+M/2].

Step A-3: reducing the sampling frequency of the filtered characteristic impedance and propagation coefficient corresponding to the full length of the transmission line to the resampling frequency f_(maxres) (f_(maxres)<f_(max)):

-   -   Calculating the number of resampling points N_(tres) in the time         domain:         N _(tres) =t _(max) ·f _(maxres)     -   Calculating the sampling time t_(new)[n₂] (n₂=1, 2, 3 . . .         N_(tres)) corresponding to the n₂ ^(th) sampling point:         t _(new)[n ₂]=(n ₂−1)/f _(maxres)     -   Obtaining the resampled sampling value by linear interpolation         method:

${z_{cres}\left\lbrack n_{2} \right\rbrack} = {{\frac{z_{c2} - z_{c1}}{t_{2} - t_{1}} \cdot \left( {{t_{new}\left\lbrack n_{2} \right\rbrack} - t_{1}} \right)} + z_{c1}}$ ${a_{res}\left\lbrack n_{2} \right\rbrack} = {{\frac{a_{2} - a_{1}}{t_{2} - t_{1}} \cdot \left( {{t_{new}\left\lbrack n_{2} \right\rbrack} - t_{1}} \right)} + a_{1}}$

Where Z_(cres) is the resampled characteristic impedance; a_(res) is the resampled propagation coefficient; t₁ and t₂ are two adjacent sampling times in the original sampling time series t, and t₁≤t_(new)[n₂]≤t₂; z_(c1) and z_(c2) are sampling values corresponding to the sampling times t₁, t₂ in the filtered characteristic impedance Z_(cfilter), respectively; a₁ and a₂ are sampling values corresponding to the sampling times t₁, t₂ in the filtered propagation coefficient a_(filter), respectively.

Step B: collecting a head end voltage value u_(m), a head end current value i_(m), a tail end voltage value u_(n), a tail end current value i_(n) of the sampling transmission line, a sampling frequency f_(s) (f_(s)<f_(maxres)), and a number of the sampling voltage and current N_(s). In order to match the voltage and current sampling rate with the propagation constant within a shorter length of the line and obtain an accurate calculation result, waveform reduction and calculation of modulus components of voltage and current are adopted in the present invention. According to the waveform reduction method, the sampling frequency is increased to the resampling f frequency, maxres and phase mode transformation is performed. The detailed steps are as follows:

Step B-1: calculating according to the waveform reduction method in Step B is as follows:

-   -   a): Setting the frequency for traveling-wave transmission         calculation as f_(maxres), inserting zero points to the original         signal x[i₂] according to the new sampling rate, i₂=1, 2, . . .         N_(s), so that the number of sampling points is changed from         N_(s) to N_(x) (N_(x)=N_(s)·N_(add), N_(add)=f_(maxres)/f_(s)),         thus obtaining a recombinant signal x_(rec)[i₃](i₃=1, 2 . . .         N_(x)):         -   x_(rec)[1+(i₂−1)·N_(dd)]=x[i₂], the values of the rest             points are zero; Where x represents the sampling values             including voltages u_(m), u_(n) and currents i_(m), i_(n) at             the sampling frequency f_(s); and x_(rec) represents the             sampling values including voltages u_(mrec), u_(nrec) and             currents i_(mrec), i_(nrec) at the sampling frequency             f_(max) after inserting the zero points;     -   b): filtering the recombinant signal x_(rec)[i₃] using the sinc         filter with Blackman window, setting a cut-off frequency         f_(c2)=1/(2·N_(add)), choosing an appropriate kernel length M₂         and a kernel h₂[j₂] of the filter, and filtering out harmonics         at higher frequencies:

${x_{restore}\left\lbrack {i_{3} + j_{2} - 1} \right\rbrack} = {\sum\limits_{j_{2} = 1}^{M_{2} + 1}{{h_{2}\left\lbrack j_{2} \right\rbrack}{x_{rec}\left\lbrack i_{3} \right\rbrack}}}$

-   -   Where j₂=1, 2 . . . M₂+1 and i₃=1, 2 . . . N_(x); x_(restore)         represents the voltage values u_(mrestore), u_(nrestore) and         currents values i_(mrestore), i_(nrestore) after waveform         reduction;     -   c): intercepting the restored voltage and current signals,         namely, removing the previous and subsequent M₂/2 sampling         points of the signal x_(restore); the restored signal is         x_(restore)[i₃+M₂/2] corresponding to the sampling time t[i₃],         the intercepted signal after waveform reduction is x_(re)[n₃]         (n₃=1, 2 . . . N_(x)−M₂/2), and x_(re)[n₃]=x_(restore)[n₃+M₂/2];         x_(re) represents the intercepted voltage values u_(mre),         u_(nre) and intercepted currents values i_(mre), i_(nre) after         waveform reduction.

Step B-2: performing phase mode transformation:

$\begin{bmatrix} u_{mmod1} \\ u_{mmod2} \end{bmatrix} = {{{\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & {- \frac{\sqrt{2}}{2}} \end{bmatrix}\begin{bmatrix} u_{mp} \\ u_{mn} \end{bmatrix}}\begin{bmatrix} u_{nmod1} \\ u_{nmod2} \end{bmatrix}} = {{{\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & {- \frac{\sqrt{2}}{2}} \end{bmatrix}\begin{bmatrix} u_{np} \\ u_{nn} \end{bmatrix}}\begin{bmatrix} i_{mmod1} \\ i_{mmod2} \end{bmatrix}} = {{{\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & {- \frac{\sqrt{2}}{2}} \end{bmatrix}\begin{bmatrix} i_{mp} \\ i_{mn} \end{bmatrix}}\begin{bmatrix} i_{nmod1} \\ i_{nmod2} \end{bmatrix}} = {\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & {- \frac{\sqrt{2}}{2}} \end{bmatrix}\begin{bmatrix} i_{np} \\ i_{nn} \end{bmatrix}}}}}$

Where the voltage values u_(mmod1), u_(nmod1) and current values i_(mmod1), i_(nmod1) are ground-mode components after waveform reduction; and voltage values u_(mmod2), u_(nmod2) and current values i_(mmod2), n_(nmod2) are differential-mode components after waveform reduction; the voltage values u_(mp), u_(mn) are p-phase and n-phase of the voltage value u_(mre), respectively; the voltage values u_(np), u_(nn) are p-phase and n-phase of the voltage value u_(nre), respectively; the current values i_(mp), i_(mn) are p-phase and n-phase of the current value i_(mre), respectively; the current values i_(np), i_(nn) are p-phase and n-phase of the current value i_(nre), respectively;

Step C: calculating differential values of voltage travelling-wave uwave_(diffm) and uwave_(diffn) at both ends, presetting a preset threshold value to u_(set), determining whether the internal fault occurs on the transmission line based on the criterion of uwave_(diffm)[n₃]≥u_(set)∥uwave_(diffn)[n₃]≥u_(set); since the present invention does not involve with the fault types, the fault identification is calculated using the voltage and current differential-mode components as follows:

Determining whether the internal fault occurs: uwave_(diffm)[n ₃]≥u _(set) ∥uwave_(diffn)[n ₃]≥u _(set)

When either of the values uwave_(diffm)[n₃] and uwave_(diffn)[n₃] is greater than the preset threshold value u_(set), it is considered that an internal fault occurs at the time t[n₃]. In order to ensure the reliability of the determination, the determination is made continuously several times or by the integral method.

The calculation of the differential values of voltage travelling-wave is as follows: uwave_(diffm)[n ₃]=(u _(mmod2)[n ₃]+z _(cres)[n ₂]*i _(mmod2)[n ₃])*a _(res)[n ₂]−(u _(nmod2)[n ₃]−z _(cres)[n ₂]*i _(nmod2)[n ₃]) uwave_(diffn)[n ₃]=(u _(nmod2)[n ₃]+z _(cres)[n ₂]*i _(nmod2)[n ₃])*a _(res)[n ₂]−(u _(mmod2)[n ₃]−z _(cres)[n ₂]*i _(mmod2)[n ₃])

Where n₃=1, 2 . . . N_(x)−M₂/2 and n₂=1, 2 . . . N_(tres); * represents the convolution calculation, the value uwave_(diffm) is the differential value between the differential-mode forward voltage traveling-wave at the head end after full-length line transmission and the differential-mode backward voltage traveling-wave at the tail end; the value uwave_(diffn) is the differential value between the differential-mode forward voltage traveling-wave at the tail end after full-length line transmission and the differential-mode backward voltage traveling-wave at the head end.

The detailed steps of the convolution calculation are as follows:

Supposing xx[j₃] is an input signal with N₁ points, j₃ is a positive integer from 1 to N₁, hh[j₄] is an impulse response with N₂ points, and j₄ is a positive integer from 1 to N₂. The convolution result of xx[j₃] and hh[j₄] is y[j₅], which is a signal with N_(t)+N₂−1 point, j₅ is a positive integer from 1 to N₁+N₂−1, then

${y\left\lbrack j_{5} \right\rbrack} = {\sum\limits_{j_{4} = 1}^{N_{2}}{h{h\left\lbrack j_{4} \right\rbrack}x{x\left\lbrack {j_{5} - j_{4}} \right\rbrack}}}$

Where j₅=1, 2 . . . N₁+N₂−1 and j₄=1, 2 . . . N₂.

The technical scheme of the present invention will be further described by the following embodiment:

Step A: calculating the characteristic impedance and propagation coefficient of the frequency-dependent transmission line in time domain; the detailed steps are as follows:

Step A-1: the VSC-HVDC transmission line is shown in FIG. 1. according to the conductance matrix G (generally ignored, can be set to 0) and the geometric transmission line parameters including the radius of conductor, the average above-ground height of the conductor, the distance between the conductor, the distance between the conductor and the mirror of another conductor, obtaining the impedance matrix of the conductor by the Carson formula, obtaining the capacitance matrix by the potential coefficient matrix, and then calculating the expression of the characteristic impedance and propagation coefficient corresponding to the full length l of the transmission line in frequency domain, and finally transforming it into the time domain by rational function fitting or inverse Fourier transformation, the detailed flow chart is shown in the FIG. 2;

Setting the sampling time t_(max), the sampling frequency f_(max), the number of sampling points N_(t) in time domain and the number of sampling points N_(f) in frequency domain as: N _(t) =N _(f) =t _(max) ·f _(max)

Calculating the sampling time t[n₁] (n₁=1, 2, 3 . . . N_(t)) corresponding to the n₁ ^(th) sampling point: t[n ₁]=(n ₁−1)/f _(max)

Calculating the frequency f[n₁] corresponding to the n₁ ^(th) sampling point: f[n ₁]=(n ₁−1)/t _(max)

-   -   a): calculating the impedance matrix Z by Carson formula:     -   Self-impedance Z_(si) (i=1,2) is:

$Z_{si} = {\left( {R_{i,{ac}} + {\Delta R_{si}}} \right) + {j\left( {{\omega\frac{\mu_{0}}{2\pi}\ln\frac{2h_{i}}{r_{i}}} + X_{i,{ac}} + {\Delta X_{si}}} \right)}}$

-   -   Where R_(i,ac) is the AC resistance of the conductor i, X_(i,ac)         is the AC reactance of the conductor i, ΔR_(si) and ΔX_(si) are         Carson's correction terms for earth return effects, h_(i) is the         average above-ground height of the conductor i, r_(i) is the         radius of the conductor i, μ₀ having the value of 4π×10⁻⁴ H/m is         the uniform permeability for both the aerial space and the         earth; and angular frequency ω=2πf[n₁];     -   Mutual impedance Z_(mij) (j=1,2 and j≠i) is:

$Z_{mij} = {Z_{mji} = {{\Delta R_{mij}} + {j\left( {{\omega\frac{\mu_{0}}{2\pi}\ln\frac{d_{{ij},{mir}}}{d_{ij}}} + {\Delta X_{mij}}} \right)}}}$

-   -   Where d_(ij) is the distance between conductor i and conductor         j, d_(ij,mir) is the distance between the conductor I and the         mirror of conductor j, ΔR_(mij) and ΔX_(mij) are Carson's         correction terms for earth return effects; and then

${Z = \begin{bmatrix} Z_{s1} & Z_{m12} \\ Z_{m21} & Z_{s2} \end{bmatrix}};$

-   -   b): calculating the potential coefficient matrix P:

${P_{si} = {\frac{1}{2\pi ɛ_{0}}\ln\frac{2h_{i}}{r_{i}}}}{P_{mij} = {P_{mji} = {\frac{1}{2\pi ɛ_{0}}\ln\frac{d_{{ij},{mir}}}{d_{ij}}}}}$

-   -   Where P_(si) is the diagonal element of matrix P, P_(mij) is the         off-diagonal element of matrix P, ε₀ is the space permittivity         value having the value of 8.85×10⁻¹² F/m, and the capacitance         matrix C=P⁻¹;     -   c): performing phase mode transformation:         Z′=SZ         C′=SC         G′=SG     -   Where Z′ is the modulus of the impedance per unit length of the         conductor, C′ is the modulus of the capacitance per unit length         of the conductor, G′ is the modulus of the conductance per unit         length of the conductor, and S is the decoupling matrix,

${S = {\frac{1}{2}\begin{bmatrix} 1 & {- 1} \\ 1 & 1 \end{bmatrix}}};$

-   -   d): calculating the function of the characteristic impedance         Z_(c) and the propagation coefficient A in frequency domain:

Z^(′) = R^(′) + j ω L^(′) ${Z_{c}(\omega)} = \sqrt{\frac{R^{\prime} + {j\;\omega\; L^{\prime}}}{G + {j\;\omega\; C^{\prime}}}}$ ${A(\omega)} = e^{{- l}\sqrt{{({R + {j\;\omega\; L^{\prime}}})}{({G + {j\;\omega\; C^{\prime}}})}}}$

-   -   Where R′ is the resistance per unit length of the conductor, L′         is the inductance per unit length of the conductor;     -   e): converting the functions from the frequency domain into the         time domain by rational function fitting method:     -   Fitting the functions Z_(c)(ω) and A(ω) in the frequency domain         by the rational function fitting method, the point where the         slope of the fitted line changes is the pole and zero point of         the rational function, so the approximate expression of the         characteristic impedance and the propagation coefficient in the         frequency domain are:

${{Z_{c,{approx}}(s)} = {k\frac{\left( {s + z_{1}} \right)\left( {s + z_{2}} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {s + z_{m1}} \right)}{\left( {s + p_{1}} \right)\left( {s + p_{2}} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {s + p_{m1}} \right)}}}{{A_{approx}(s)} = {e^{{- s}\;\tau_{\min}}k\frac{\left( {s + z_{1}} \right)\left( {s + z_{2}} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {s + z_{m1}} \right)}{\left( {s + p_{1}} \right)\left( {s + p_{2}} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {s + p_{m2}} \right)}}}$

-   -   Where s=jω and m1<m2; Z_(mi) is the zero point, m1=1, 2, 3 . . .         , P_(m2) is the pole, m2=1, 2, 3 . . . , the zero and pole         points are both negative real numbers; k is coefficient, τ_(min)         is the shortest propagation time of the wave; Z_(c,approx)(s) is         the rational function approximation of characteristic impedance,         A_(approx)(s) is the rational function approximation of         propagation coefficient;     -   Converting the functions into the time domain:

     Z_(C)(t) = k₀δ(t) + k₁e^(−p₁t) + k₂e^(−p₂t)+  …   + k_(m1)e^(−p_(m1)t) ${a(t)} = \left\{ \begin{matrix} {{{k_{1}e^{- {p_{1}{({t - \tau_{\min}})}}}} + {k_{2}e^{- {p_{2}{({t - \tau_{\min}})}}}} + \mspace{14mu}\ldots\mspace{14mu} + {k_{m\; 2}e^{- {p_{m\; 2}{({t - \tau_{\min}})}}}}},} & {{{if}\mspace{14mu} t} \geq \tau_{\min}} \\ {0,} & {{{if}\mspace{14mu} t} < \tau_{\min}} \end{matrix} \right.$

-   -   Where Z_(c)(t) is the time domain expression of the         characteristic impedance, a(t) is the time domain expression of         the propagation coefficient, k_(m2), is the coefficient of the         expanded rational function; the characteristic impedance         Z_(c)[n₁] and the propagation coefficient and corresponding to         each sampling time t[n₁] can be achieved;

Step A-2: in order to match the calculated characteristic impedance and propagation coefficient with the collected voltage and current in frequency domain, filtering the characteristic impedance and propagation coefficient using a sinc filter with a Blackman window;

-   -   Calculating the kernel h[j₁] of a sinc filter with Blackman         window:

$\quad\left\{ \begin{matrix} {{h\left\lbrack j_{1} \right\rbrack} = {K\frac{\sin\left( {2\pi\;{f_{c}\left( {j_{1} - {M/2}} \right)}} \right.}{j_{1} - {M/2}}{w\left\lbrack j_{1} \right\rbrack}}} & {j_{1} \neq {M/2}} \\ {{h\left\lbrack j_{1} \right\rbrack} = {2\pi\; f_{c}}} & {j_{1} = {M/2}} \end{matrix} \right.$

-   -   Performing normalization:

${h\left\lbrack j_{1} \right\rbrack} = \frac{h\left\lbrack j_{1} \right\rbrack}{\underset{i = 1}{\sum\limits^{M + 1}}{h\left\lbrack j_{1} \right\rbrack}}$

-   -   Where j₁=1, 2 . . . M+1, f_(c) is the cut-off frequency having         the value within the range of 0 to 0.5; if the transmission line         is short but the sampling frequency f_(max) is high, when         performing convolution calculation with voltage and current         values, it is required to reduce the sampling frequency to         f_(maxres), so f_(c)=1/(2×(f_(max)/f_(maxres)+1)). Wherein M is         the kernel length of the filter, which should be an even number         and corresponding to the cut-off frequency, which can have the         value of 4f_(c); K is the coefficient; w[j₁] is the function of         Blackman window and is expressed as follows:         w[j ₁]=0.42−0.5 cos(2πj ₁ /M)+0.08 cos(4πj ₁ /M)     -   Performing filtering:

${{z_{cfilter}\left\lbrack {n_{1} + j_{1} - 1} \right\rbrack} = {\sum\limits_{j_{1} = 1}^{M + 1}{{h\left\lbrack j_{1} \right\rbrack}{z_{c}\left\lbrack n_{1} \right\rbrack}}}},{{a_{filter}\left\lbrack {n_{1} + j_{1} - 1} \right\rbrack} = {\sum\limits_{j_{1} = 1}^{M + 1}{{h\left\lbrack j_{1} \right\rbrack}{a\left\lbrack n_{1} \right\rbrack}}}}$

-   -   Where n₁=1, 2 . . . N_(t); Z_(cfilter) is the filtered         characteristic impedance; a_(filter) is the filtered propagation         coefficient; the filtered signal has a delay of M/2 point         compared with the original signal, so the filtered point         corresponding to the sampling time t[n₁] is Z_(cfilter)[n₁+M/2]         and a_(filter)[n₁+M/2].

Step A-3: reducing the sampling frequency of the filtered characteristic impedance and propagation coefficient corresponding to the full length of the transmission line to the resampling frequency f_(maxres) (f_(maxres)<f_(max)):

-   -   Calculating the number of resampling points N_(tres) in the time         domain:         N _(tres) =t _(max) ·f _(maxres)     -   Calculating the sampling time t_(new)[n₂] (n₂=1, 2, 3 . . .         N_(tres)) corresponding to the n₂ ^(th) sampling point:         t _(new)[n ₂]=(n ₂−1)/f _(maxres)     -   Obtaining the resampled sampling value by linear interpolation         method:

${z_{cres}\left\lbrack n_{2} \right\rbrack} = {{{\frac{z_{c2} - z_{c1}}{t_{2} - t_{1}} \cdot \left( {{t_{new}\left\lbrack n_{2} \right\rbrack} - t_{1}} \right)} + {z_{c1}{a_{res}\left\lbrack n_{2} \right\rbrack}}} = {{\frac{a_{2} - a_{1}}{t_{2} - t_{1}} \cdot \left( {{t_{new}\left\lbrack n_{2} \right\rbrack} - t_{1}} \right)} + a_{1}}}$

Where Z_(cres) is the resampled characteristic impedance; a_(res) is the resampled propagation coefficient; t₁ and t₂ are two adjacent sampling times in the original sampling time series t, and t₁≤t_(new)[n₂]≤t₂; z_(c1) and z_(c2) are sampling values corresponding to the sampling times t₁, t₂ in the filtered characteristic impedance Z_(cfilter), respectively; a₁ and a₂ are sampling values corresponding to the sampling times t₁, t₂ in the filtered propagation coefficient a_(filter), respectively.

Step B: collecting a head end voltage value u_(m), a head end current value i_(m), a tail end voltage value u_(n), a tail end current value i_(n) of the sampling transmission line, a sampling frequency f_(s) (f_(s)(f_(s)<f_(maxres)), and a number of the sampling voltage and current N_(s). In order to match the voltage and current sampling rate with the propagation constant within a shorter length of the line and obtain an accurate calculation result, waveform reduction and calculation of modulus components of voltage and current are adopted in the present invention. According to the waveform reduction method, the sampling frequency is increased to a resampling frequency f_(maxres) and phase mode transformation is performed. The detailed steps are as follows:

Step B-1: as shown in FIG. 3, calculating according to the waveform reduction method in Step B is as follows:

-   -   a): Setting the frequency for traveling-wave transmission         calculation as f_(maxres), inserting zero points to the original         signal x[i₂] according to the new sampling rate, i₂=1, 2, . . .         N so that the number of sampling points is changed from N_(s) to         N_(x) (=N_(s)·N_(add), N_(add)=f_(maxres)/f_(s)), thus obtaining         a recombinant signal x_(rec)[i₃] (i₃=1, 2 . . . N_(x)):         -   x_(rec)[1+(i₂−1)·N_(dd)]=x[i₂], the values of the rest             points are zero;     -   Where x represents the sampling values including voltages u_(m),         u_(n) and currents i_(m), i_(n) at the sampling frequency f_(s);         and x_(ree) represents the sampling values including voltages         u_(mrec), and currents i_(mrec), i_(nrec) at the sampling         frequency f_(max) after inserting the zero points;     -   b): filtering the recombinant signal x_(rec)[i₃] using the sinc         filter with Blackman window, setting a cut-off frequency         f_(c2)=1/(2·N_(add)), choosing an appropriate kernel length M₂         and a kernel h₂[j₂] of the filter, and filtering out harmonics         at higher frequencies:

${x_{restore}\left\lbrack {i_{3} + j_{2} - 1} \right\rbrack} = {\sum\limits_{j_{2} = 1}^{M_{2} + 1}{{h_{2}\left\lbrack j_{2} \right\rbrack}{x_{rec}\left\lbrack i_{3} \right\rbrack}}}$

-   -   Where j₂=1, 2 . . . M₂+1 and i₃=1, 2 . . . N_(x); x_(restore)         represents the voltage values u_(mrestore), u_(nrestore) and         currents values i_(mrestore), i_(nrestore) after waveform         reduction;     -   c): intercepting the restored voltage and current signals,         namely, removing the previous and subsequent M₂/2 sampling         points of the signal x_(restore); the restored signal is         x_(restore)[i₃+M₂/2] corresponding to the sampling time t[i₃],         the intercepted signal after waveform reduction is x_(re)[n₃]         (n₃=1, 2 . . . N_(x)−M₂/2), and x_(re)[n₃]=x_(restore)[n₃+M₂/2];         x_(re) represents the intercepted voltage values u_(mre),         u_(nre) and intercepted currents values i_(mre), i_(nre) after         waveform reduction.

Step B-2: performing phase mode transformation:

$\begin{bmatrix} u_{m\;{mod}\; 1} \\ u_{m\;{mod}\; 2} \end{bmatrix} = {{{\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & {- \frac{\sqrt{2}}{2}} \end{bmatrix}\begin{bmatrix} u_{mp} \\ u_{mn} \end{bmatrix}}\begin{bmatrix} u_{n\;{mod}\; 1} \\ u_{n\;{mod}\; 2} \end{bmatrix}} = {{{\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & {- \frac{\sqrt{2}}{2}} \end{bmatrix}\begin{bmatrix} u_{np} \\ u_{nn} \end{bmatrix}}\begin{bmatrix} i_{m\;{mod}\; 1} \\ i_{m\;{mod}\; 2} \end{bmatrix}} = {{{\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & {- \frac{\sqrt{2}}{2}} \end{bmatrix}\begin{bmatrix} i_{mp} \\ i_{mn} \end{bmatrix}}\begin{bmatrix} i_{n\;{mod}\; 1} \\ i_{n\;{mod}\; 2} \end{bmatrix}} = {\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & {- \frac{\sqrt{2}}{2}} \end{bmatrix}\begin{bmatrix} i_{np} \\ i_{nn} \end{bmatrix}}}}}$

Where the voltage values u_(mmod1), u_(nmod1) and current values i_(mmod1), i_(nmod1) are ground-mode components after waveform reduction; and voltage values u_(mmod2), u_(nmod2) and current values i_(mmod2), i_(nmod2) are differential-mode components after waveform reduction; the voltage values u_(mp), u_(mp) are p-phase and n-phase of the voltage value u_(mre), respectively; the voltage values u_(np), u_(nn) are p-phase and n-phase of the voltage value u_(nre), respectively; the current values i_(mp), i_(mn) are p-phase and n-phase of the current value i_(mre), respectively; the current values i_(np), i_(nn) are p-phase and n-phase of the current value i_(nre), respectively;

Step C: calculating the differential values of voltage travelling-wave uwave_(diffm) and uwave_(diffn) at both ends, presetting a preset threshold value u_(set), determining whether the internal fault occurs on the transmission line based on the criterion of uwave_(diffm)[n₃]≥u_(set)∥uwave_(diffn)[n₃]≥u_(set); since the present invention does not involved with the fault types, the fault identification is calculated using the voltage and current differential-mode components as follows:

Determining whether the internal fault occurs: uwave_(diffm)[n ₃]≥u _(set) ∥uwave_(diffn)[n ₃]≥u _(set)

When either of the values uwave_(diffm)[n₃] and uwave_(diffn)[n₃] is greater than the preset threshold value u_(set), it is considered that the internal fault occurs at a time t[n₃]. In order to ensure the reliability of the determination, the determination is made continuously several times or by the integral method.

The calculation of the differential values of voltage travelling-wave is as follows: uwave_(diffm)[n ₃]=u _(mfw)[n ₃]*a _(res)[n ₂]−u _(nbw)[n ₃] uwave_(diffn)[n ₃]=u _(nfw)[n ₃]*a _(res)[n ₂]−u _(mbw)[n ₃]

Where the value uwave_(diffm) is the differential value between the differential-mode forward voltage traveling-wave at the head end after full-length line transmission and the differential-mode backward voltage traveling-wave at the tail end; the value uwave_(diffn) is the differential value between the differential-mode forward voltage traveling-wave at the tail end after full-length line transmission and the differential-mode backward voltage traveling-wave at the head end;

The calculation of the voltage travelling-wave is as follows: u _(mfw)[n ₃]u _(mmod2)[n ₃]+z _(cres)[n ₂]*i _(mmod2)[n ₃] u _(nfw)[n ₃]=u _(nmod2)[n ₃]+z _(cres)[n ₂]*i _(nmod2)[n ₃] u _(mbw)[n ₃]=u _(mmod2)[n ₃]+z _(cres)[n ₂]*i _(mmod2)[n ₃] u _(nbw)[n ₃]=u _(nmod2)[n ₃]−z _(cres)[n ₂]*i _(nmod2)[n ₃]

Where n₃=1, 2 . . . N_(x)−M₂/2 and n₂=1, 2 . . . N_(tres); * represents the convolution calculation, u_(mfw) is a differential-mode forward voltage traveling-wave at the head end, u_(mbw) is the differential-mode backward voltage traveling-wave at the head end, u_(nfw) is a differential-mode forward voltage traveling-wave at the tail end, u_(nbw) is the differential-mode backward voltage traveling-wave at the tail end.

The detailed steps of the convolution calculation are as follows:

Supposing xx[j₃] is an input signal with N₁ points, j₃ is a positive integer from 1 to N₁, hh[j₄] is an impulse response with N₂ points, and j₄ is a positive integer from 1 to N₂. The convolution result of xx[j₃] and hh[j₄] is y[j₅], which is a signal with N₁+N₂−1 points, j₅ is a positive integer from 1 to N₁+N₂−1, then

${y\left\lbrack j_{5} \right\rbrack} = {\sum\limits_{j_{4} = 1}^{N_{2}}{{{hh}\left\lbrack j_{4} \right\rbrack}{{xx}\left\lbrack {j_{5} - j_{4}} \right\rbrack}}}$

Where j₅=1, 2 . . . N₁+N₂−1 and j₄=1, 2 . . . N₂. Only the first N_(x) signals are adopted during the calculation of the voltage traveling-wave differential values.

Although the functions and working processes of the present invention have been described above with reference to the accompanying drawings, the present invention is not limited thereto. The foregoing specific implementations are merely illustrative but not limiting. A person of ordinary skill in the art may make various forms under the teaching of the present invention without departing from the purpose of the present invention and the protection scope of the appended claims, and all the forms shall fall into the protection scope of the present invention. 

What is claimed is:
 1. A calculation method for travelling-wave differential protection of VSC-HVDC transmission lines, considering the influence of the frequency-related parameters, comprising the following steps: calculating a characteristic impedance and a propagation coefficient of a frequency-dependent transmission line in time domain; collecting voltage and current signals at the both ends of the transmission line and then calculating a forward and a backward voltage travelling-wave, respectively; comparing a differential value of voltage travelling-wave with a preset threshold value so as to determine whether an internal fault occurs; wherein the method comprises detailed steps as follows: Step A: calculating the characteristic impedance and propagation coefficient of the frequency-dependent transmission line in time domain; Step B: collecting the voltage and current signals at the both ends of the transmission line, comprising a head end voltage value u_(m), a head end current value i_(m), a tail end voltage value u_(n), a tail end current value i_(n) of the transmission line, a sampling frequency f_(s), and a number of the sampling voltage and current N_(s); according to the waveform reduction method, filtering and intercepting the voltage and current signals using a filter with kernel length of M₂ and then increasing the sampling frequency f_(s) to a resampling frequency f_(maxres) (f_(maxres)>f_(s)), and the number of sampling points becoming N_(x)−M₂/2, wherein N_(x)=N_(s)*f_(maxres)/f_(s); and performing phase mode transformation; Step C: comparing a differential value of voltage travelling-wave with a preset voltage threshold value so as to determine whether the internal fault occurs: calculating differential values of voltage travelling-wave uwave_(diffm) and uwave_(dffn) at both ends, presetting a preset threshold value u_(set), determining whether the internal fault occurs on the transmission line based on the criterion of uwave_(diffm)[n₃≥u_(set)∥uwave_(diffn)[n₃]≥u_(set) (n₃=1, 2 . . . N_(x)−M₂/2 differential-mode components of voltage and current to realize fault recognition; further, determining whether the internal fault occurs: uwave_(diffm)[n ₃]≥u _(set) ∥uwave_(diffn)[n ₃]≥u _(set) when either of uwave_(diffm)[n₃] or uwave_(diffn)[n₃] is greater than the preset threshold value u_(set), it is considered that the internal fault occurs at a time t[n₃]; in order to ensure the reliability of the determination, the determination is made continuously several times or by the integral method.
 2. The calculation method for travelling-wave differential protection of VSC-HVDC transmission lines according to claim 1, wherein the detailed steps of Step A are as follows: Step A-1: according to a conductance matrix G and geometric transmission line parameters including a radius of conductor, an average above-ground height of the conductor, a distance between the conductors, a distance between the conductor and the mirror of another conductor, obtaining an impedance matrix of the conductor by the Carson formula, obtaining a capacitance matrix using a potential coefficient matrix, and then calculating the expression of the characteristic impedance and propagation coefficient corresponding to a full length/of the transmission line in frequency domain, and finally transforming it into the time domain by rational function fitting or inverse Fourier transformation; Step A-2: in order to match the calculated characteristic impedance and propagation coefficient with the collected voltage and current in frequency domain, filtering the characteristic impedance and propagation coefficient using a sinc filter with a Blackman window; Step A-3: reducing the sampling frequency of the filtered characteristic impedance and propagation coefficient corresponding to the full length of the transmission line to to the resampling frequency f_(maxres).
 3. The calculation method for travelling-wave differential protection of VSC-HVDC transmission lines according to claim 2, wherein the detailed steps of Step A-1 are as follows: Setting a sampling time t_(max), a sampling frequency f_(max), a number of sampling points N_(t) in time domain and a number of sampling points N_(f) in frequency domain as: N _(t) =N _(f) =t _(max) ·f _(max) Calculating a sampling time t[n₁] corresponding to the n₁ ^(th) sampling point: t[n ₁]=(n ₁−1)/f _(max) n₁=1, 2, 3 . . . N_(t), calculating a frequency f[n₁] corresponding to the n₁ ^(th) sampling point: f[n ₁]=(n ₁−1)/t _(max) a): calculating an impedance matrix Z using Carson formula: Self-impedance Z_(si) (i=1,2) is: $Z_{si} = {\left( {R_{i,{ac}} + {\Delta\; R_{si}}} \right) + {j\left( {{\omega\frac{\mu_{0}}{2\pi}\ln\frac{2h_{i}}{r_{i}}} + X_{i,{ac}} + {\Delta\; X_{si}}} \right)}}$ Where R_(i,ac) is an AC resistance of a conductor i, X_(i,ac) is an AC reactance of the conductor i, ΔR_(si) and ΔX_(si) are Carson's correction terms for earth return effects, h_(i) is the average above-ground height of the conductor i, r_(i) is the radius of the conductor i, μ₀ having the value of 4π×10⁻⁴ H/m is the uniform permeability for both the aerial space and the earth and angular frequency ω=2πf[n₁]; Mutual impedance Z_(mij) is: $Z_{mij} = {Z_{mji} = {{\Delta R_{mij}} + {j\left( {{\omega\frac{\mu_{0}}{2\pi}\frac{d_{{ij},{mir}}}{d_{ij}}} + {\Delta\; X_{mij}}} \right)}}}$ Where j=1,2 and j≠i, d_(ij) is a distance between conductor i and conductor j, d_(ij,mir) is a distance between the conductor i and the mirror of the conductor j, ΔR_(mij) and ΔX_(mij) are Carson's correction terms for earth return effects; and then ${z = \begin{bmatrix} Z_{s1} & Z_{m12} \\ Z_{m\; 21} & Z_{s2} \end{bmatrix}};$ b): calculating a potential coefficient matrix P: $P_{si} = {\frac{1}{2{\pi ɛ}_{0}}\ln\frac{2h_{i}}{r_{i}}}$ $P_{mij} = {P_{mji} = {\frac{1}{2{\pi ɛ}_{0}}\ln\frac{d_{{ij},{mir}}}{d_{ij}}}}$ Where P_(si) is a diagonal element of the matrix P, P_(mij) is a off-diagonal element of the matrix P, ε₀ is a space permittivity value having the value of 8.85×10⁻¹² F/m, and a capacitance matrix C=P⁻¹; c): performing phase mode transformation: Z′=SZS ⁻¹ C′=SCS ⁻¹ G′=SGS ⁻¹ Where Z′ is a modulus of the impedance per unit length of the conductor, C′ is a modulus of the capacitance per unit length of the conductor, G′ is a modulus of the conductance per unit length of the conductor, and S is a decoupling matrix, ${S = {\frac{1}{2}\begin{bmatrix} 1 & 1 \\ 1 & {­1} \end{bmatrix}}};$ d): calculating a function of a characteristic impedance Z_(c) and a propagation coefficient A in frequency domain: Z^(′) = R^(′) + j ω L^(′) ${Z_{c}(\omega)} = \sqrt{\frac{R^{\prime} + {j\;\omega\; L^{\prime}}}{G^{\prime} + {j\;\omega\; C^{\prime}}}}$ ${A(\omega)} = e^{{- l}\sqrt{{({R^{\prime} + {j\;\omega\; L}})}{({G^{\prime} + {j\;\omega\; C}})}}}$ Where R′ is a resistance per unit length of the conductor, L′ is a inductance per unit length of the conductor; e): converting the functions from the frequency domain into the time domain by rational function fitting method: Fitting functions Z_(c)(ω) and A(ω) in the frequency domain by the rational function fitting method, a point where the slope of the fitted line changes is a pole and a zero point of the rational function, so the approximate expression of the characteristic impedance and the propagation coefficient in the frequency domain are: ${Z_{c,{approx}}(s)} = {k\frac{\left( {s + z_{1}} \right)\left( {s + z_{2}} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {s + z_{m\; 1}} \right)}{\left( {s + p_{1}} \right)\left( {s + p_{2}} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {s + p_{m\; 1}} \right)}}$ ${A_{approx}(s)} = {e^{{- s}\;\tau_{\min}}k\frac{\left( {s + z_{1}} \right)\left( {s + z_{2}} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {s + z_{m\; 1}} \right)}{\left( {s + p_{1}} \right)\left( {s + p_{2}} \right)\mspace{14mu}\ldots\mspace{14mu}\left( {s + p_{m\; 2}} \right)}}$ Where s=jω and m1<m2; Z_(mi) is the zero point, m1=1, 2, 3 . . . , P_(m2) is the pole, m2=1, 2, 3 . . . , the zero and pole points are both negative real numbers; k is a coefficient, τ_(min) is the shortest propagation time of the wave; Z_(c,approx)(S) is the rational function approximation of characteristic impedance, A_(approx)(S) is the rational function approximation of propagation coefficient; Converting the functions into the time domain:   Z_(o)(t) = k₀δ(t) + k₁e^(−p₂t) + k₂e^(−p₂t)+  …   + k_(m 1)e^(−p_(m 1)t) ${a(t)} = {\quad\left\{ \begin{matrix} {{{k_{1}e^{- {p_{1}{({t - \tau_{\min}})}}}} + {k_{2}e^{- {p_{2}{({t - \tau_{\min}})}}}} + \mspace{14mu}\ldots\mspace{14mu} + {k_{m\; 2}e^{- {p_{m\; 2}{({t - \tau_{\min}})}}}}},} & {{{if}\mspace{14mu} t} \geq \tau_{\min}} \\ 0 & {{{if}\mspace{14mu} t} < \tau_{\min}} \end{matrix} \right.}$ Where Z_(c)(t) is a time domain expression of the characteristic impedance, a(t) is a time domain expression of the propagation coefficient, k_(m2) is a coefficient of the expanded rational function; a characteristic impedance Z_(b)[n₁] and a propagation coefficient a[n₁] corresponding to each sampling time t[n₁] can be achieved.
 4. The calculation method for travelling-wave differential protection of VSC-HVDC transmission lines according to claim 2, wherein the detailed steps of Step A-2 are as follows: Calculating the kernel h[j₁] of a sinc filter with Blackman window: $\quad\left\{ \begin{matrix} {{h\left\lbrack j_{1} \right\rbrack} = {K\frac{\sin\left( {2\pi\;{f_{c}\left( {j_{1} - {M/2}} \right)}} \right.}{j_{1} - {M/2}}{w\left\lbrack j_{1} \right\rbrack}}} & \left( {j_{1} \neq {M/2}} \right) \\ {{h\left\lbrack j_{1} \right\rbrack} = {2\pi\; f_{c}}} & \left( {j_{1} = {M/2}} \right) \end{matrix} \right.$ Performing normalization: ${h\left\lbrack j_{1} \right\rbrack} = \frac{h\left\lbrack j_{1} \right\rbrack}{\underset{i = 1}{\sum\limits^{M + 1}}{h\left\lbrack j_{1} \right\rbrack}}$ Where j₁=1, 2 . . . M+1, f_(c) is a cut-off frequency having the value within the range of 0 to 0.5; M is a kernel length of the filter, which should be an even number; K is a coefficient; w[j₁] is the function of Blackman window and is expressed as follows: w[j ₁]=0.42−0.5 cos(2πj ₁ /M)+0.08 cos(4πj ₁ /M) Performing Filtering ${{Z_{cfilter}\left\lbrack {n_{1} + j_{1} - 1} \right\rbrack} = \left. {\sum\limits_{{j_{1}}_{\;^{= 1}}}^{M + 1}{h\left\lbrack j_{1} \right\rbrack}}\rightarrow{z_{c}\left\lbrack n_{1} \right\rbrack} \right.},{{a_{filter}\left\lbrack {n_{1} + j_{1} - 1} \right\rbrack} = {\underset{\;^{j_{1} = 1}}{\sum\limits^{M + 1}}{{h\left\lbrack j_{1} \right\rbrack}{a\left\lbrack n_{1} \right\rbrack}}}}$ Where n₁=1, 2 . . . N_(t); Z_(cfilter) is a filtered characteristic impedance; a_(filter) is a filtered propagation coefficient; the filtered signal has a delay of M/2 point compared with the original signal, so the filtered point corresponding to the sampling time t[n₁] is Z_(cfilter)[n¹+M/2] and a_(filter)[n₁+M/2].
 5. The calculation method for travelling-wave differential protection of VSC-HVDC transmission lines according to claim 2, wherein the detailed steps of Step A-3 are as follows: Calculating number of resampling points N_(tres) in the time domain: N _(tres) =t _(max) ·f _(maxres) Where f_(maxres) is the resampled sampling frequency; Calculating the sampling time t_(new)[n₂] corresponding to the n₂ ^(th) sampling point: t _(new)[n ₂]=(n ₂−1)/f _(maxres) Where n₂=1, 2, 3 . . . N_(tres); Obtaining a resampled sampling value by linear interpolation method: ${z_{cres}\left\lbrack n_{2} \right\rbrack} = {{\frac{z_{c\; 2} - z_{c\; 1}}{t_{2} - t_{1}} \cdot \left( {{t_{new}\left\lbrack n_{2} \right\rbrack} - t_{1}} \right)} + z_{c\; 1}}$ ${a_{res}\left\lbrack n_{2} \right\rbrack} = {{\frac{a_{2} - a_{1}}{t_{2} - t_{1}} \cdot \left( {{t_{new}\left\lbrack n_{2}\  \right\rbrack} - t_{1}} \right)} + a_{1}}$ Where Z_(cres) a the resampled characteristic impedance; a_(res) is a resampled propagation coefficient; t₁ and t₂ are two adjacent sampling times in an original sampling time series t, and t₁≤t_(new)[n₂]≤t₂; z_(c1) and z_(c2) are sampling values corresponding to the sampling times t₁, t₂ in the filtered characteristic impedance Z_(cfilter), respectively; a₁ and a₂ are sampling values corresponding to the sampling times t₁, t₂ in the filtered propagation coefficient a_(filter), respectively.
 6. The calculation method for travelling-wave differential protection of VSC-HVDC transmission lines according to claim 1, wherein the detailed steps of calculating according to the waveform reduction method in the Step B are as follows: a): Setting the frequency for traveling-wave transmission calculation as f_(maxres), inserting zero points to an original signal x[i₂] according to the new sampling rate, i₂=1, 2, . . . N_(s), so that the number of sampling points is changed from N_(s) to N_(x), N_(x)=N_(s)·N_(add), N_(add)=f_(maxres)/f_(s), thus obtaining a recombinant signal x_(rec)[i₃]: x_(rec)[1+(i₂−1)·N_(dd)]=x[i₂], the values of the rest points are zero; Where i₃=1, 2 . . . N_(x), x represents sampling values including voltages u_(m), u_(n) and currents i_(m), i_(n) at the sampling frequency f_(s); and x_(rec) represents sampling values including voltages u_(mrec), u_(nrec) and currents i_(mrec), i_(nrec) at the sampling frequency f_(max) after inserting the zero points; b): filtering the recombinant signal x_(rec)[i₃] using the sinc filter with Blackman window, setting a cut-off frequency f_(c2)=1/(2·N_(add)), choosing an appropriate kernel length M₂ and a kernel h₂[j₂] of the filter, and filtering out harmonics at higher frequencies: ${x_{restore}\left\lbrack {i_{3} + j_{2} - 1} \right\rbrack} = {\sum\limits_{j_{2} = 1}^{M_{2} + 1}{{h_{2}\left\lbrack j_{2} \right\rbrack}{x_{rec}\left\lbrack i_{3} \right\rbrack}}}$ Where j₂=1, 2 . . . M₂+1 and i₃=1, 2 . . . N_(x); X_(restore) represents voltage values u_(mrestore), u_(nrestore) and currents values i_(mrestore), i_(nrestore) after waveform reduction; c): intercepting the restored voltage and current signals, namely, removing the previous and subsequent M₂/2 sampling points of the signal X_(restore); the restored signal is x_(restore)[i₃+M₂/2] corresponding to the sampling time t[i₃], the intercepted signal after waveform reduction is x_(re)[n₃] (n₃=1, 2 . . . N_(x)−M₂/2), and x_(re)[n₃]=x_(restore)[n₃+M₂/2]; represents intercepted voltage values u_(mre), u_(nre) and intercepted currents values i_(mre), i_(nre) after waveform reduction.
 7. The calculation method for travelling-wave differential protection of VSC-HVDC transmission lines according to claim 1, wherein the detailed steps of performing phase mode transformation in the Step B are as follows: $\begin{bmatrix} u_{m\;{mod}\; 1} \\ u_{m\;{mod}\; 2} \end{bmatrix} = {{{\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & {\text{-}\frac{\sqrt{2}}{2}} \end{bmatrix}\begin{bmatrix} u_{mp} \\ u_{mn} \end{bmatrix}}\begin{bmatrix} u_{n\;{mod}\; 1} \\ u_{n\;{mod}\; 2} \end{bmatrix}} = {{{\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & {\text{-}\frac{\sqrt{2}}{2}} \end{bmatrix}\begin{bmatrix} u_{np} \\ u_{nn} \end{bmatrix}}\begin{bmatrix} i_{m\;{mod}\; 1} \\ i_{m\;{mod}\; 2} \end{bmatrix}} = {{{\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & {\text{-}\frac{\sqrt{2}}{2}} \end{bmatrix}\begin{bmatrix} i_{mp} \\ i_{mn} \end{bmatrix}}\begin{bmatrix} i_{n\;{mod}\; 1} \\ i_{n\;{mod}\; 2} \end{bmatrix}} = {\begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & {\text{-}\frac{\sqrt{2}}{2}} \end{bmatrix}\begin{bmatrix} i_{np} \\ i_{nn} \end{bmatrix}}}}}$ Where voltage values u_(mmod1), u_(nmod1) and current values i_(mmod1), i_(nmod1) are ground-mode components after waveform reduction; and voltage values u_(mmod2), u_(nmod2) and current values i_(mmod2), i_(nmod2) are differential-mode components after waveform reduction; voltage values u_(mp), u_(mn) are p-phase and n-phase of the voltage value u_(mre), respectively; voltage values u_(np), u_(nn) are p-phase and n-phase of the voltage value u_(nre), respectively; current values i_(mp), i_(mn) are p-phase and n-phase of the current value i_(mre), respectively; current values i_(np), i_(nn) are p-phase and n-phase of the current value i_(nre), respectively.
 8. The calculation method for travelling-wave differential protection of VSC-HVDC transmission lines according to claim 1, wherein the detailed steps of Step C are as follows: Calculating the differential values of voltage travelling-wave at both ends: uwave_(diffm)[n ₃]=(u _(mmod2)[n ₃]+z _(cres)[n ₂]*i _(mmod2)[n ₃])*a _(res)[n ₂]−(u _(nmod2)[n ₃]−z _(cres)[n ₂]*i _(nmod2)[n ₃]) uwave_(diffn)[n ₃]=(u _(nmod2)[n ₃]+z _(cres)[n ₂]*i _(nmod2)[n ₃])*a _(res)[n ₂]−(u _(mmod2)[n ₃]−z _(cres)[n ₂]*i _(mmod2)[n ₃]) Where n₃=1 , 2 . . . N_(x)−M₂/2 and n₂=1, 2 . . . N_(tres); * represents the convolution calculation, the value uwave_(diffm) is the differential value between the differential-mode forward voltage traveling-wave at the head end after full-length line transmission and the differential-mode backward voltage traveling-wave at the tail end; the value uwave_(diffn) is the differential value between the differential-mode forward voltage traveling-wave at the tail end after full-length line transmission and the differential-mode backward voltage traveling-wave at the head end; Calculating the convolution calculation: Supposing xx[j₃] is an input signal with N₁ points, j₃ is a positive integer from 1 to N₁, hh[j₄] is an impulse response with N₂ points, and j₄ is a positive integer from 1 to N₂; the convolution result of xx[j₃] and hh[j₄] is y[j₅], which is a signal with N₁+N₂−1 points, j₅ is a positive integer from 1 to N₁+N₂−1, then ${y\left\lbrack j_{5} \right\rbrack} = {\sum\limits_{j_{4} = 1}^{N_{2}}{{{hh}\left\lbrack j_{4} \right\rbrack}{{xx}\left\lbrack {j_{5} - j_{4}} \right\rbrack}}}$ Where j₅=1, 2 . . . N₁+N₂−1 and j₄=1, 2 . . . N₂. 